Let's break down the problem step by step.

Part (a)

We are given two equations:

1. $y = 2x$
2. $y = 2x + 1$

(a)(i) Gradient of the line $y = 2x + 1$

The general form of a linear equation is $y = mx + c$, where:

• $m$ is the gradient (slope),
• $c$ is the y-intercept.

For the equation $y = 2x + 1$, we can directly see that the gradient $m = 2$.

Answer (a)(i): The gradient is 2.

(a)(ii) Solve $2x + 1 = 2x$

We need to find the value of $x$ such that $x > 0$ and $2x + 1 = 2x$.

1. Subtract $2x$ from both sides: $2x + 1 - 2x = 2x - 2x$ This simplifies to: $1 = 0$

This is a contradiction, which means there is no value of $x$ that satisfies the equation $2x + 1 = 2x$ when $x > 0$.

Answer (a)(ii): There is no solution for $x > 0$.

Part (b)

We are given the equation $y = k a^x$, and we need to find the values of $k$ and $a$.

(b)(i) Value of $k$

$k$ represents the initial value when $x = 0$. Based on the diagram (which I assume shows an exponential graph), the value of $y$ when $x = 0$ is the constant $k$. From the general form $y = k a^x$, substitute $x = 0$: $y = k a^0 = k(1) = k$ So, $y(0) = k$, which is the y-intercept of the graph.

You would typically find this value from the graph (not given in the problem), but it’s the initial value when $x = 0$.

Answer (b)(i): The value of $k$ depends on the y-intercept, which can be read off from the graph.

(b)(ii) Value of $a$

$a$ is the base of the exponential function, which can be determined by observing how the graph increases. If the graph's shape shows rapid growth, $a > 1$, and if it shows decay, $0 < a < 1$. The exact value of $a$ also needs to be read off the graph based on the nature of the curve.

Answer (b)(ii): The value of $a$ depends on the growth rate of the exponential function, which is read from the graph.

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Would you like me to go into more details on any part of this explanation or help with another example?

5 related questions:

1. What is the difference between a gradient and y-intercept in linear equations?
2. How do we identify contradictions in algebraic equations?
3. How does the value of $a$ affect the shape of an exponential graph?
4. What happens to an exponential graph when $k$ is negative?
5. Can the equation $2x + 1 = 2x$ have a solution if we remove the constraint $x > 0$?

Tip:

When solving linear equations, always check for contradictions or undefined results when simplifying!